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Contents 1 Introduction 2 Simulation of a weather radar signal from a fixed range 2.1 Simulation of a weather radar signal from a fixed direction 3 Conclusion 4 Acknowledgements 5 References

Introduction

In weather radar experiments, the objective is to measure reflectivity, velocity and spectral width of rain for different ranges. Several papers have demonstrated methods for solving these three parameters with different kind of coding schemes and analysis algorithms, see for example [Pirttilä,Sachi]. The analysis methods are based on computing the so-called autocorrelation function for different ranges r from the radar. The parameters are then deduced with some spectral methods.

radarsim

Simulation of a weather radar signal from a fixed range

Because the power spectrum S(ω) is a Gaussian shaped function, it is symmetric with respect to some reference point ω₀. With the help of the convolution, we can represent the power spectrum as follows.

Because autocorrelation function and power spectrum are a Fourier transform pair, we can denote the absolute value of the autocorrelation function | C(t) | = conv(c(t),c(t)).

Let us now consider how to construct the radar signal. Let us define a signal as follows.

We denote by N(0,1) a Gaussian distribution with zero mean and variance 1. Now let us consider the time series

The autocorrelation function of $\xi(t)$ can be represented as follows.

Now we can continue the formulation

Fourier transformation of the complex white noise process can be given as . We can assume, that the Fourier transformation of c is real. Then we will get

= 2conv(c(t),c(t)) = 2 | C(t) |

Thus by dividing the random vector ξ(t) by will result the wanted time-series given some certain absolute value of autocorrelation function.

This is first part of the generation of the autocorrelation function. Often the autocorrelation functions are modulated by some exponential function. In equation \ref{autocorr}, we gave the autocorrelation for a fixed range. It it clear, that it can be also represented as follows.

C(t) = exp(iω₀t) | C(t) |

It can be easily shown, that we will get the wanted behaviour by considering the following time-series

Now let us verify this. Let us denote and . Now the autocorrelation function turns out to be

= exp(iω₀t) | C(t) |

This verifies the proper properties of equation (\ref{keijo}).

Due to the properties of the convolution, the length of the radar signal ξ can be as long as wanted. Thus the time space simulation is more flexible than the frequency space simulation. However, if the power spectrum turns out to be very complex, it might be easier to make the simulation in the frequency space.

Simulation of a weather radar signal from a fixed direction

Conclusion

Acknowledgements

This work has been partially funded by Academy of Finland.

References

[1] J Pirttilä, A Huuskonen and M Lehtinen, Solving the Range-Doppler Dilemma with Ambiguity-Free Measurements Developed for Incoherent Scatter Radars, Cost 75 Advanced weather radar systems international seminar, Locarno Switzerland 1999.

[2] J Pirttilä, M Lehtinen, A Huuskonen and M Markkanen, A Proposed Solution to the Range-Doppler Dilemma of Weather Radar Measurements by Using SMPRF Codes, Practical Results and a Comparison to Operational Measurements, {\it J. Appl. Meteor.} 2004